A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Sufficient conditions to specify a stochastic process?

In order to specify a stochastic process, is it sufficient to specify all the finite-dimensional distributions of the stochastic process? Can there exist two stochastic processes that agree in all the ...
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Proving that $Z_t=g(X_t)$ is a martingale if and only if $\mathbb{E}(Z_t|Z_s)=Z_s \ \forall t>s \geq 0$. ($X_t$ Markov)

I want to prove the next property: Let $X_t$ be a Markov process (so $\mathbb{E}(X_t|\mathcal{F}_s)=\mathbb{E}(X_t|X_s)$ where $\mathcal{F_t}=\sigma(X_s, s\leq t)$). Suppose that $Z_t=g(X_t)$ where ...
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Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
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12 views

Motivation behind the definition of the Itô integral for elementary predictable processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$. A real-valued stochastic process $H=(H_t)_{t\ge 0}$ is called elementary ...
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Proving technique used to show an equivalence to the definition of a Markov process

Let $X=(X_t)_{t\in I}$ be Markov process with values in a Polish space $E$. I want to show, that there exists a stochastic kernel $\kappa:E\times\mathcal{B}(E)^{\otimes I}\to [0,1]$ such that ...
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Random process question. How to find the mean,autocorrelation and WSS? [on hold]

A fellow student posted this question on our whatsapp group for the course. A random process $$X(t) = A\cos(\omega t)B\sin(\omega t) $$ where $A$ and $B$ are random variables. Find the ...
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1answer
22 views

Solving the Geometric Brownian Motion on a general interval.

I know that the Geometric Brownian Motion, with the expression $dX_t = v X_t dt + \sigma X_t dW_t$ has the next solution $$X_t = X_0 e^{\sigma W_t+ (v-\frac{\sigma ^2}{2})t}$$ on the interval $[0,T]$. ...
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1answer
27 views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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19 views

A property of the space of rcll functions

Let $E$ be a locally compact separable metric space, $$\Omega=\{f:[0,\infty]\to E\mid f \text{ is rcll}\}$$ and $P$ be a probability on $\Omega$ which equiped the $\sigma$-algebra generated by ...
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32 views

Find the expected value .

I am having a trouble in in the following questions. I would like if some one could help me. Thank you so much for your time. Simplify $$K=\int_0^T\left[\int_0^t e^{-\alpha(t-s)}dJ(s)\right] dt $$ ...
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1answer
25 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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46 views

Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
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1answer
26 views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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23 views

Compound Poisson Process property:$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq n)=\mathbb{P}(\sum^{N_{t_4}}_{i=N_{t_3}+1}J_i \leq n)$

I am trying to demostrate that the Compound Poisson Process has independent increments, and I have a problen because I have to use that: :$$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq ...
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Is the Martingale property still true for $\xi$ not necessarily $C^1$?

Denote $$M(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$ Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ the ...
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Understanding Quadratic Variation

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much ...
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Reference request for this topics

I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience.    1.2. Stationary measure.    1.3. ...
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Non existence of probabilty measures.

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ ist standard Wiener. This solution is ...
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how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
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24 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
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Bounded L2 increments for an Ornstein Uhlenbeck type process

Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlennbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can ...
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Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
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2answers
48 views

Proving that the Poisson compound process has independent increments

Let $X_t=\sum_{i=1}^{N_t}J_i$ be a compound Poisson Process, where $J_i$ are independent and equidistributed. I have to prove that for every $0<t_1<t_2 \leq t_3<t_4$ : $X_{t_4}-X_{t_3}$ is ...
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Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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1answer
35 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
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35 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
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1answer
25 views

Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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Transient discrete time Markov chain on integers: can direction of flow be proven?

I'm not very familiar with the theory of Markov chains, and I'd like to learn how complicated the following problem actually is. Let there be a discrete time Markov chain on $\mathbb{Z}$, where the ...
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1answer
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Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
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Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
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Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
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Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
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35 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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1answer
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Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
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39 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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Aggregate arrivals from a renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process". The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF $f(t)$. ...
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1answer
29 views

Aggregate arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=λe^{−λt},~t>0$. ($λ$ is the arrival rate of the ...
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3answers
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Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
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Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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14 views

process stochastics and branching process [duplicate]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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1answer
27 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
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8 views

discontinuous Gaussian field

I am trying to build an example of a discontinuous Gaussian field. The simplest I could come up with is the following: Let $Y,Z$ be two independent brownian motions on $[0,1]$, and $T$ a uniform ...
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22 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
3
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1answer
57 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...